A precise estimate of the rate of convergence in the Central Limit Theorem in Hilbert space

Let
$$ S_n=n^{-1/2}\sigma^{-1}\sum_1^n(X_i-\mathbf EX_i),\quad\sigma^2=\mathbf E|X_1-\mathbf EX_1|^2, $$
be the normed sum of independent identically distributed random variables $X_i$ with values in a separable Hilbert space $H$. Denote by $V$ the covariance operator of $X$, and let $Y$ be an $H$-valued $(0,\sigma^{-2}V)$ Gaussian random variable. The authors prove that there exist an absolute constant such that for any $a\in H$ and $r\geqslant0$
$$ |\mathbf P(|S_n-a|<r)-\mathbf P(|Y-a|<r)|\leqslant c\biggl(\prod_1^6\sigma_i^{-1}\biggr)\sigma^3\mathbf E|X_1-\mathbf EX_1|^3(1+|a|^3)n^{-1/2}, $$
where $\sigma_1^2\geqslant\sigma_2^2\geqslant\dotsb$ are the eigenvalues of $V$. Up to the value of $c$, this estimate is unimprovable in general.
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